Cross-constrained variational methods for the nonlinear Klein-Gordon equations with an inverse square potential

In this paper, we apply a cross-constrained variational approach for the nonlinear Klein-Gordon equations with an inverse square potential in three space dimensions (which is a representative of the class of equations of interest) based on the relationship between a type of cross-constrained variational problem and energy. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we first derive a sharp threshold for global existence and blow-up of solutions to the Cauchy problem for the equations under study. On the other hand, we get an answer of the question: how small are the initial data, the global solutions exist?